Calculus Examples

$y = x^{4} - 3 x + 2$

Step 1

Set $y$ as a function of $x$ .

$f (x) = x^{4} - 3 x + 2$

Step 2

Find the derivative.

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Step 2.1

Differentiate.

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Step 2.1.1

By the Sum Rule, the derivative of $x^{4} - 3 x + 2$ with respect to $x$ is $\frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 3 x] + \frac{d}{d x} [2]$ .

$\frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 3 x] + \frac{d}{d x} [2]$

Step 2.1.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$4 x^{3} + \frac{d}{d x} [- 3 x] + \frac{d}{d x} [2]$

Step 2.2

Evaluate $\frac{d}{d x} [- 3 x]$ .

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Step 2.2.1

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3 x$ with respect to $x$ is $- 3 \frac{d}{d x} [x]$ .

$4 x^{3} - 3 \frac{d}{d x} [x] + \frac{d}{d x} [2]$

Step 2.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$4 x^{3} - 3 \cdot 1 + \frac{d}{d x} [2]$

Step 2.2.3

Multiply $- 3$ by $1$ .

$4 x^{3} - 3 + \frac{d}{d x} [2]$

Step 2.3

Differentiate using the Constant Rule.

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Step 2.3.1

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$4 x^{3} - 3 + 0$

Step 2.3.2

Add $4 x^{3} - 3$ and $0$ .

$4 x^{3} - 3$

Step 3

Set the derivative equal to $0$ then solve the equation $4 x^{3} - 3 = 0$ .

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Step 3.1

Add $3$ to both sides of the equation.

$4 x^{3} = 3$

Step 3.2

Divide each term in $4 x^{3} = 3$ by $4$ and simplify.

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Step 3.2.1

Divide each term in $4 x^{3} = 3$ by $4$ .

$\frac{4 x^{3}}{4} = \frac{3}{4}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $4$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{4 x^{3}}{4} = \frac{3}{4}$

Step 3.2.2.1.2

Divide $x^{3}$ by $1$ .

$x^{3} = \frac{3}{4}$

Step 3.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \sqrt[3]{\frac{3}{4}}$

Step 3.4

Simplify $\sqrt[3]{\frac{3}{4}}$ .

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Step 3.4.1

Rewrite $\sqrt[3]{\frac{3}{4}}$ as $\frac{\sqrt[3]{3}}{\sqrt[3]{4}}$ .

$x = \frac{\sqrt[3]{3}}{\sqrt[3]{4}}$

Step 3.4.2

Multiply $\frac{\sqrt[3]{3}}{\sqrt[3]{4}}$ by $\frac{{\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{2}}$ .

$x = \frac{\sqrt[3]{3}}{\sqrt[3]{4}} \cdot \frac{{\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{2}}$

Step 3.4.3

Combine and simplify the denominator.

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Step 3.4.3.1

Multiply $\frac{\sqrt[3]{3}}{\sqrt[3]{4}}$ by $\frac{{\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{2}}$ .

$x = \frac{\sqrt[3]{3} {\sqrt[3]{4}}^{2}}{\sqrt[3]{4} {\sqrt[3]{4}}^{2}}$

Step 3.4.3.2

Raise $\sqrt[3]{4}$ to the power of $1$ .

$x = \frac{\sqrt[3]{3} {\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{1} {\sqrt[3]{4}}^{2}}$

Step 3.4.3.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = \frac{\sqrt[3]{3} {\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{1 + 2}}$

Step 3.4.3.4

Add $1$ and $2$ .

$x = \frac{\sqrt[3]{3} {\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{3}}$

Step 3.4.3.5

Rewrite ${\sqrt[3]{4}}^{3}$ as $4$ .

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Step 3.4.3.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{4}$ as $4^{\frac{1}{3}}$ .

$x = \frac{\sqrt[3]{3} {\sqrt[3]{4}}^{2}}{{(4^{\frac{1}{3}})}^{3}}$

Step 3.4.3.5.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x = \frac{\sqrt[3]{3} {\sqrt[3]{4}}^{2}}{4^{\frac{1}{3} \cdot 3}}$

Step 3.4.3.5.3

Combine $\frac{1}{3}$ and $3$ .

$x = \frac{\sqrt[3]{3} {\sqrt[3]{4}}^{2}}{4^{\frac{3}{3}}}$

Step 3.4.3.5.4

Cancel the common factor of $3$ .

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Step 3.4.3.5.4.1

Cancel the common factor.

$x = \frac{\sqrt[3]{3} {\sqrt[3]{4}}^{2}}{4^{\frac{3}{3}}}$

Step 3.4.3.5.4.2

Rewrite the expression.

$x = \frac{\sqrt[3]{3} {\sqrt[3]{4}}^{2}}{4^{1}}$

Step 3.4.3.5.5

Evaluate the exponent.

$x = \frac{\sqrt[3]{3} {\sqrt[3]{4}}^{2}}{4}$

Step 3.4.4

Simplify the numerator.

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Step 3.4.4.1

Rewrite ${\sqrt[3]{4}}^{2}$ as $\sqrt[3]{4^{2}}$ .

$x = \frac{\sqrt[3]{3} \sqrt[3]{4^{2}}}{4}$

Step 3.4.4.2

Raise $4$ to the power of $2$ .

$x = \frac{\sqrt[3]{3} \sqrt[3]{16}}{4}$

Step 3.4.4.3

Rewrite $16$ as $2^{3} \cdot 2$ .

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Step 3.4.4.3.1

Factor $8$ out of $16$ .

$x = \frac{\sqrt[3]{3} \sqrt[3]{8 (2)}}{4}$

Step 3.4.4.3.2

Rewrite $8$ as $2^{3}$ .

$x = \frac{\sqrt[3]{3} \sqrt[3]{2^{3} \cdot 2}}{4}$

Step 3.4.4.4

Pull terms out from under the radical.

$x = \frac{\sqrt[3]{3} \cdot 2 \sqrt[3]{2}}{4}$

Step 3.4.4.5

Combine exponents.

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Step 3.4.4.5.1

Combine using the product rule for radicals.

$x = \frac{2 \sqrt[3]{3 \cdot 2}}{4}$

Step 3.4.4.5.2

Multiply $3$ by $2$ .

$x = \frac{2 \sqrt[3]{6}}{4}$

Step 3.4.5

Cancel the common factor of $2$ and $4$ .

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Step 3.4.5.1

Factor $2$ out of $2 \sqrt[3]{6}$ .

$x = \frac{2 (\sqrt[3]{6})}{4}$

Step 3.4.5.2

Cancel the common factors.

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Step 3.4.5.2.1

Factor $2$ out of $4$ .

$x = \frac{2 \sqrt[3]{6}}{2 \cdot 2}$

Step 3.4.5.2.2

Cancel the common factor.

$x = \frac{2 \sqrt[3]{6}}{2 \cdot 2}$

Step 3.4.5.2.3

Rewrite the expression.

$x = \frac{\sqrt[3]{6}}{2}$

Step 4

Solve the original function $f (x) = x^{4} - 3 x + 2$ at $x = \frac{\sqrt[3]{6}}{2}$ .

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Step 4.1

Replace the variable $x$ with $\frac{\sqrt[3]{6}}{2}$ in the expression.

$f (\frac{\sqrt[3]{6}}{2}) = {(\frac{\sqrt[3]{6}}{2})}^{4} - 3 \frac{\sqrt[3]{6}}{2} + 2$

Step 4.2

Simplify the result.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Apply the product rule to $\frac{\sqrt[3]{6}}{2}$ .

$f (\frac{\sqrt[3]{6}}{2}) = \frac{{\sqrt[3]{6}}^{4}}{2^{4}} - 3 \frac{\sqrt[3]{6}}{2} + 2$

Step 4.2.1.2

Simplify the numerator.

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Step 4.2.1.2.1

Rewrite ${\sqrt[3]{6}}^{4}$ as $\sqrt[3]{6^{4}}$ .

$f (\frac{\sqrt[3]{6}}{2}) = \frac{\sqrt[3]{6^{4}}}{2^{4}} - 3 \frac{\sqrt[3]{6}}{2} + 2$

Step 4.2.1.2.2

Raise $6$ to the power of $4$ .

$f (\frac{\sqrt[3]{6}}{2}) = \frac{\sqrt[3]{1296}}{2^{4}} - 3 \frac{\sqrt[3]{6}}{2} + 2$

Step 4.2.1.2.3

Rewrite $1296$ as $6^{3} \cdot 6$ .

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Step 4.2.1.2.3.1

Factor $216$ out of $1296$ .

$f (\frac{\sqrt[3]{6}}{2}) = \frac{\sqrt[3]{216 (6)}}{2^{4}} - 3 \frac{\sqrt[3]{6}}{2} + 2$

Step 4.2.1.2.3.2

Rewrite $216$ as $6^{3}$ .

$f (\frac{\sqrt[3]{6}}{2}) = \frac{\sqrt[3]{6^{3} \cdot 6}}{2^{4}} - 3 \frac{\sqrt[3]{6}}{2} + 2$

Step 4.2.1.2.4

Pull terms out from under the radical.

$f (\frac{\sqrt[3]{6}}{2}) = \frac{6 \sqrt[3]{6}}{2^{4}} - 3 \frac{\sqrt[3]{6}}{2} + 2$

Step 4.2.1.3

Raise $2$ to the power of $4$ .

$f (\frac{\sqrt[3]{6}}{2}) = \frac{6 \sqrt[3]{6}}{16} - 3 \frac{\sqrt[3]{6}}{2} + 2$

Step 4.2.1.4

Cancel the common factor of $6$ and $16$ .

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Step 4.2.1.4.1

Factor $2$ out of $6 \sqrt[3]{6}$ .

$f (\frac{\sqrt[3]{6}}{2}) = \frac{2 (3 \sqrt[3]{6})}{16} - 3 \frac{\sqrt[3]{6}}{2} + 2$

Step 4.2.1.4.2

Cancel the common factors.

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Step 4.2.1.4.2.1

Factor $2$ out of $16$ .

$f (\frac{\sqrt[3]{6}}{2}) = \frac{2 (3 \sqrt[3]{6})}{2 (8)} - 3 \frac{\sqrt[3]{6}}{2} + 2$

Step 4.2.1.4.2.2

Cancel the common factor.

$f (\frac{\sqrt[3]{6}}{2}) = \frac{2 (3 \sqrt[3]{6})}{2 \cdot 8} - 3 \frac{\sqrt[3]{6}}{2} + 2$

Step 4.2.1.4.2.3

Rewrite the expression.

$f (\frac{\sqrt[3]{6}}{2}) = \frac{3 \sqrt[3]{6}}{8} - 3 \frac{\sqrt[3]{6}}{2} + 2$

Step 4.2.1.5

Combine $- 3$ and $\frac{\sqrt[3]{6}}{2}$ .

$f (\frac{\sqrt[3]{6}}{2}) = \frac{3 \sqrt[3]{6}}{8} + \frac{- 3 \sqrt[3]{6}}{2} + 2$

Step 4.2.1.6

Move the negative in front of the fraction.

$f (\frac{\sqrt[3]{6}}{2}) = \frac{3 \sqrt[3]{6}}{8} - \frac{3 \sqrt[3]{6}}{2} + 2$

Step 4.2.2

To write $- \frac{3 \sqrt[3]{6}}{2}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$f (\frac{\sqrt[3]{6}}{2}) = \frac{3 \sqrt[3]{6}}{8} - \frac{3 \sqrt[3]{6}}{2} \cdot \frac{4}{4} + 2$

Step 4.2.3

Write each expression with a common denominator of $8$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.2.3.1

Multiply $\frac{3 \sqrt[3]{6}}{2}$ by $\frac{4}{4}$ .

$f (\frac{\sqrt[3]{6}}{2}) = \frac{3 \sqrt[3]{6}}{8} - \frac{3 \sqrt[3]{6} \cdot 4}{2 \cdot 4} + 2$

Step 4.2.3.2

Multiply $2$ by $4$ .

$f (\frac{\sqrt[3]{6}}{2}) = \frac{3 \sqrt[3]{6}}{8} - \frac{3 \sqrt[3]{6} \cdot 4}{8} + 2$

Step 4.2.4

Combine the numerators over the common denominator.

$f (\frac{\sqrt[3]{6}}{2}) = \frac{3 \sqrt[3]{6} - 3 \sqrt[3]{6} \cdot 4}{8} + 2$

Step 4.2.5

Simplify each term.

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Step 4.2.5.1

Simplify the numerator.

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Step 4.2.5.1.1

Multiply $4$ by $- 3$ .

$f (\frac{\sqrt[3]{6}}{2}) = \frac{3 \sqrt[3]{6} - 12 \sqrt[3]{6}}{8} + 2$

Step 4.2.5.1.2

Subtract $12 \sqrt[3]{6}$ from $3 \sqrt[3]{6}$ .

$f (\frac{\sqrt[3]{6}}{2}) = \frac{- 9 \sqrt[3]{6}}{8} + 2$

Step 4.2.5.2

Move the negative in front of the fraction.

$f (\frac{\sqrt[3]{6}}{2}) = - \frac{9 \sqrt[3]{6}}{8} + 2$

Step 4.2.6

The final answer is $- \frac{9 \sqrt[3]{6}}{8} + 2$ .

$- \frac{9 \sqrt[3]{6}}{8} + 2$

Step 5

The horizontal tangent line on function $f (x) = x^{4} - 3 x + 2$ is $y = - \frac{9 \sqrt[3]{6}}{8} + 2$ .

$y = - \frac{9 \sqrt[3]{6}}{8} + 2$

Step 6

The result can be shown in multiple forms.

Exact Form:

$y = - \frac{9 \sqrt[3]{6}}{8} + 2$

Decimal Form:

$y = - 0.04426066 \dots$

Step 7